Question

A woman leaves home and walks 7 miles west, then 4 miles southwest. How far from home is she? (Round your answer to three decimal places.)

In direction north of east must she walk to head directly home? (Round your answer to three decimal places.)

Answer 1

The woman is approximately 8.062 miles from home. She needs to walk approximately 29.744° north of east to head directly home.

Step-by-step explanation:

To find the distance from home, we can imagine the woman's movements as two separate legs of a right triangle. The first leg is 7 miles to the west, and the second leg is 4 miles to the southwest. These legs form a right angle at the starting point. Using the Pythagorean theorem, we can find the hypotenuse of the triangle, which represents the distance from home.

Solving for the hypotenuse (distance from home):

Distance from home = √((7 miles)^2 + (4 miles)^2)

Distance from home ≈ 8.062 miles

To find the direction north of east that the woman must walk to head directly home, we can use trigonometry. We know the lengths of the two legs of the right triangle and can find the angle formed between the hypotenuse and the east direction. This angle represents the direction the woman should walk.

Solving for the angle:

Angle = arctan(4 miles / 7 miles)

Angle ≈ 29.744°